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Theorem issubrg2 20494
Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubrg2.b 𝐡 = (Baseβ€˜π‘…)
issubrg2.o 1 = (1rβ€˜π‘…)
issubrg2.t Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
issubrg2 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝑅,𝑦   π‘₯, Β· ,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   1 (π‘₯,𝑦)

Proof of Theorem issubrg2
Dummy variables 𝑣 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 20479 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 ∈ (SubGrpβ€˜π‘…))
2 issubrg2.o . . . 4 1 = (1rβ€˜π‘…)
32subrg1cl 20482 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 1 ∈ 𝐴)
4 issubrg2.t . . . . . 6 Β· = (.rβ€˜π‘…)
54subrgmcl 20486 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)
653expb 1117 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)
76ralrimivva 3194 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)
81, 3, 73jca 1125 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴))
9 simpl 482 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 𝑅 ∈ Ring)
10 simpr1 1191 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 𝐴 ∈ (SubGrpβ€˜π‘…))
11 eqid 2726 . . . . . . 7 (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs 𝐴)
1211subgbas 19057 . . . . . 6 (𝐴 ∈ (SubGrpβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜(𝑅 β†Ύs 𝐴)))
1310, 12syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 𝐴 = (Baseβ€˜(𝑅 β†Ύs 𝐴)))
14 eqid 2726 . . . . . . 7 (+gβ€˜π‘…) = (+gβ€˜π‘…)
1511, 14ressplusg 17244 . . . . . 6 (𝐴 ∈ (SubGrpβ€˜π‘…) β†’ (+gβ€˜π‘…) = (+gβ€˜(𝑅 β†Ύs 𝐴)))
1610, 15syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (+gβ€˜π‘…) = (+gβ€˜(𝑅 β†Ύs 𝐴)))
1711, 4ressmulr 17261 . . . . . 6 (𝐴 ∈ (SubGrpβ€˜π‘…) β†’ Β· = (.rβ€˜(𝑅 β†Ύs 𝐴)))
1810, 17syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ Β· = (.rβ€˜(𝑅 β†Ύs 𝐴)))
1911subggrp 19056 . . . . . 6 (𝐴 ∈ (SubGrpβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
2010, 19syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
21 simpr3 1193 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)
22 oveq1 7412 . . . . . . . . 9 (π‘₯ = 𝑒 β†’ (π‘₯ Β· 𝑦) = (𝑒 Β· 𝑦))
2322eleq1d 2812 . . . . . . . 8 (π‘₯ = 𝑒 β†’ ((π‘₯ Β· 𝑦) ∈ 𝐴 ↔ (𝑒 Β· 𝑦) ∈ 𝐴))
24 oveq2 7413 . . . . . . . . 9 (𝑦 = 𝑣 β†’ (𝑒 Β· 𝑦) = (𝑒 Β· 𝑣))
2524eleq1d 2812 . . . . . . . 8 (𝑦 = 𝑣 β†’ ((𝑒 Β· 𝑦) ∈ 𝐴 ↔ (𝑒 Β· 𝑣) ∈ 𝐴))
2623, 25rspc2v 3617 . . . . . . 7 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴 β†’ (𝑒 Β· 𝑣) ∈ 𝐴))
2721, 26syl5com 31 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒 Β· 𝑣) ∈ 𝐴))
28273impib 1113 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒 Β· 𝑣) ∈ 𝐴)
29 issubrg2.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘…)
3029subgss 19054 . . . . . . . . . 10 (𝐴 ∈ (SubGrpβ€˜π‘…) β†’ 𝐴 βŠ† 𝐡)
3110, 30syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 𝐴 βŠ† 𝐡)
3231sseld 3976 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝑒 ∈ 𝐴 β†’ 𝑒 ∈ 𝐡))
3331sseld 3976 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝑣 ∈ 𝐴 β†’ 𝑣 ∈ 𝐡))
3431sseld 3976 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 ∈ 𝐡))
3532, 33, 343anim123d 1439 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) β†’ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)))
3635imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡))
3729, 4ringass 20158 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ ((𝑒 Β· 𝑣) Β· 𝑀) = (𝑒 Β· (𝑣 Β· 𝑀)))
3837adantlr 712 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ ((𝑒 Β· 𝑣) Β· 𝑀) = (𝑒 Β· (𝑣 Β· 𝑀)))
3936, 38syldan 590 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ ((𝑒 Β· 𝑣) Β· 𝑀) = (𝑒 Β· (𝑣 Β· 𝑀)))
4029, 14, 4ringdi 20163 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ (𝑒 Β· (𝑣(+gβ€˜π‘…)𝑀)) = ((𝑒 Β· 𝑣)(+gβ€˜π‘…)(𝑒 Β· 𝑀)))
4140adantlr 712 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ (𝑒 Β· (𝑣(+gβ€˜π‘…)𝑀)) = ((𝑒 Β· 𝑣)(+gβ€˜π‘…)(𝑒 Β· 𝑀)))
4236, 41syldan 590 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (𝑒 Β· (𝑣(+gβ€˜π‘…)𝑀)) = ((𝑒 Β· 𝑣)(+gβ€˜π‘…)(𝑒 Β· 𝑀)))
4329, 14, 4ringdir 20164 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ ((𝑒(+gβ€˜π‘…)𝑣) Β· 𝑀) = ((𝑒 Β· 𝑀)(+gβ€˜π‘…)(𝑣 Β· 𝑀)))
4443adantlr 712 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐡 ∧ 𝑣 ∈ 𝐡 ∧ 𝑀 ∈ 𝐡)) β†’ ((𝑒(+gβ€˜π‘…)𝑣) Β· 𝑀) = ((𝑒 Β· 𝑀)(+gβ€˜π‘…)(𝑣 Β· 𝑀)))
4536, 44syldan 590 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ ((𝑒(+gβ€˜π‘…)𝑣) Β· 𝑀) = ((𝑒 Β· 𝑀)(+gβ€˜π‘…)(𝑣 Β· 𝑀)))
46 simpr2 1192 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 1 ∈ 𝐴)
4732imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ 𝐡)
4829, 4, 2ringlidm 20168 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑒 ∈ 𝐡) β†’ ( 1 Β· 𝑒) = 𝑒)
4948adantlr 712 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐡) β†’ ( 1 Β· 𝑒) = 𝑒)
5047, 49syldan 590 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ ( 1 Β· 𝑒) = 𝑒)
5129, 4, 2ringridm 20169 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑒 ∈ 𝐡) β†’ (𝑒 Β· 1 ) = 𝑒)
5251adantlr 712 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐡) β†’ (𝑒 Β· 1 ) = 𝑒)
5347, 52syldan 590 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (𝑒 Β· 1 ) = 𝑒)
5413, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53isringd 20190 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝑅 β†Ύs 𝐴) ∈ Ring)
5531, 46jca 511 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴))
5629, 2issubrg 20473 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴)))
579, 54, 55, 56syl21anbrc 1341 . . 3 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)) β†’ 𝐴 ∈ (SubRingβ€˜π‘…))
5857ex 412 . 2 (𝑅 ∈ Ring β†’ ((𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴) β†’ 𝐴 ∈ (SubRingβ€˜π‘…)))
598, 58impbid2 225 1 (𝑅 ∈ Ring β†’ (𝐴 ∈ (SubRingβ€˜π‘…) ↔ (𝐴 ∈ (SubGrpβ€˜π‘…) ∧ 1 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ Β· 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  +gcplusg 17206  .rcmulr 17207  Grpcgrp 18863  SubGrpcsubg 19047  1rcur 20086  Ringcrg 20138  SubRingcsubrg 20469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-subg 19050  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-subrng 20446  df-subrg 20471
This theorem is referenced by:  opprsubrg  20495  subrgint  20497  issubrg3  20502  issubrgd  21045  cnsubrglem  21310  cnsubrglemOLD  21311  mplsubrg  21906  mplind  21973  dmatsrng  22358  scmatsrng  22377  scmatsrng1  22380  cpmatsrgpmat  22578
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